Conclusion

In these thesis a rheology was investigated, where the viscosity not only depends on temper-ature and depth but also on a damage parameter. Temporal evolution of the damage was described by an additional equation featuring source, sink and advection of damage.

The aim of the new rheology was to improve the plate{like behavior of the lithospheric mate-rial in numerical simulations. In particular it was expected to (1) produce narrow spreading centers and low viscosity zones (LVZs) above downwellings and therefore piecewise uniform surface velocities, (2) increase the asymmetry of subduction zones and (3) create long living passive fault structures.

Systematical parameter studies for dierent convective regimes demonstrated that some of these expectations were indeed fullled.

This rheology is able to develop sharp and self{focused passive spreading centers and LVZs at downwellings. In consequence highly uniform surface velocities are obtained.

It can produce signicant asymmetry in both, the velocity of the two plates at a subduction zone and the penetration of the subducted material in the mantle.

In addition, the simulations demonstrate the importance of a depth{dependent viscos-ity.

Although these features show that the rheology employed here is signicantly more success-ful in reproducing plate{like behavior than previously published models, some important features of Earth's plate tectonics are still not explained appropriately:

A complete one-side subduction observed on Earth is not found here.

If signicant asymmetry is obtained in simulations the length of the corresponding plates is often too small and the subduction angle is sometimes wrong.

126

The subduction of spreading centers is not possible.

Estimations for the damage source aEarth and sink bEarth needed to break a stagnant lid on Earth are signicantly too high.

Long living passive faults are not obtained.

Nevertheless, in my opinion this kind of damage{, temperature{ and depth{dependent rhe-ology has a good potential for further explaining plate tectonics on Earth. The results from Bercovici (1996,1998) about the evolution of transform faults and Tackley (2000c) support these expectations. However, the introduction of further geophysical aspects like the elas-ticity of plates or the brittle breaking process for the upper 10 or 20 km of the lithosphere might be advisable.

127

128

Appendix A

A.1 Non dimensional quantities in the Hydrodynamic equations

The following section describes how the dimensional quantities in the hydrodynamic equations are non-dimensionalized. The dimensional quantities are marked with the~sign.

For the bottom heated system we use:

x~= _h^x z~= ^z_h

~t = ^t_h²⁰ T~= _T^T1^;T0

;T⁰

d~= _d^d0

p~= ^(p;g0^00z)h2

~= 0

~= 0

~= 0

x and z are the horizontal and vertical spatial coordinates, t is the time,

T the temperature, d the inverse grain size, p the pressure,

the viscosity

the thermal diusivity ~the density.

g is the gravitational acceleration, the thermal expansivity,

⁰ the density at the surface h the height of the Earth mantle,

⁰ the constant thermal diusivity in the box 129

⁰ the viscosity at the surface and for a non-dimensional damage of one.

T⁰ is the surface temperature T¹ the bottom temperature and d⁰ a typical grain size at the surface.

Typical values for scaling parameters in the Earth's mantle are:

Quantity Meaning Value for the Unit

Earth's mantle

g gravity acceleration 10 ms^;2

thermal expansivity 210⁵ K^;1

T^1;T⁰ temperature dierence 3000 K

between top and bottom of the box

d⁰ Typical inverse grain 10³ m^;1

size at the surface

q rate of internal heat generation 510^;9 W m^;3 s^;1 per unit volume and time

cp specic heat 1:2510³ W kg^;1K^;1

Table A.1: Values for the scaling parameters in the Earth's mantle

The Rayleigh number obtained from the non{dimensionalization of the hydrodynamic equa-tions for a bottom heated model is:

Rabh= g⁰(T^1;T⁰)h³

^{00 (A.1)}

For an internally heated system I use

q~=q=q⁰ to dene the non-dimensional temperature as T~= ^cp_h^0(2T_q0^;T0), where

q is the rate of internal heating per unit volume and unit time and cp the specic heat.

The Rayleigh number for an internally heated system is therefore:

Raih= gq⁰h⁵

²⁰⁰cp: (A.2)

130

Acknowledgments

First at all, I wish to thank Prof. Dr. Ulrich Christensen for supervising these thesis.

Prof. Dr. David Bercovici made one fantastic year at the University of Hawaii pos-sible. A special thank to him for many scientic suggestions as well as for teaching me how to become really cynical.

The professors Tilgner, Lube, Werner, Glatzel and Kneer most kindly accepted the great honor to listen to the defense of these thesis.

Johannes Wicht, Georg Kaufmann, Ursula Wuehlner and Ajay Manglik signicantly improved the structure of this work.

Another "thank you" to all the kind people in the Institute of Geophysics who al-ways believed (and hoped) that I will nish my PhD some day.

Clearly, I would not have survived especially the last few weeks without the help of Janio and my family. Thanks for the great support.

My PhD was also supported by scholarships from the University of Goettingen, the German federal state Niedersachsen, the DAAD, the University of Hawaii and the Graduiertenkolleg "Stroemungsinstabilitaeten". Tanks a lot to all of them.

131

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136

Lebenslauf

28.06.1971 geboren in Essen, deutscher Staatsangehoeriger Schule:

1977-1981 Besuch der Grundschule Essen{Burgaltendorf 1981-1990 Schueler des humanistischen Burggymnasiums Essen 31.05.1990 Abitur

Zivildienst:

01.07.1990 Beginn des Zivildienstes auf der Pegestation des antroposophischen Bettina{von{Arnim{Altenheimes in Essen{Stadtwald

Studium:

01.10.1991 Aufnahme des Studiums der Physik und der Mathematik an der Georg{August{Universitaet Goettingen

18.10.1993 Vordiplom Physik 22.04.1994 Vordiplom Mathematik

01.10.1995 Beginn der Diplomarbeit bei Prof. Dr. U. Christensen am Institut fuer Geophysik der Universitaet Goettingen

Thema: "Ein Mehrgitterverfahren zur Loesung zweidimensionaler Konvektionsprobleme mit variabler Viskositaet"

27.06.1997 Diplom Physik

Job:01.09.1997 Mitarbeit im magnetotellurischen und seismologischen Eifel{Plume{Projekt der Universitaet Goettingen Promotion:

01.04.1998 Beginn der Promotion bei Prof. Dr. U. Christensen

Thema: "Plate tectonics in computational simulations of terrestrial mantle convection with grain{size{dependent rheology"

09.01.2000 Beginn eines einjaehrigen Aufenthaltes an der

University of Hawaii in Honolulu im Rahmen der Promotion 10.07.2000 Teilnahme an einer Summerschool ueber Parallelcomputing

im NASA Goddard Space Flight Center 31.12.2001 Voraussichtliches Ende der Promotion

Im Dokument Plate tectonics in computational simulations of terrestrial mantle convection with grain-size-dependent rheology (Seite 132-143)